Statistical Physics, Second Revised and Enlarged Edition

Amodel example 7

avalid distribution must satisfythe two conditionsimpliedbythe macrostate
∑

j

nnj=N (1.2)

∑

j

nnjεεj=U (1.3)

Equation (1.2) ensures that the distribution contains the correct number of particles.
Equation (1.3)followsfrom (1.1), andguarantees thatdistribution correspondstothe
correct value ofU.All of the conditions of the(N,U,V)macrostate are now taken
care of.

1 .5.3 Countingmicrostates

Next we needto count up the number ofmicrostates consistent witheachvalidset
of distribution numbers. Usually, and especiallyfor a largesystem, each distribution
{nnj}will be associated with a very large number of microstates. This number we
callt({nnj}).Thedependence ofton{nnj}isapure combinatorialproblem. The result
is very different for an assembly of localized particles (in which the particles are
distinguishablebytheirlocality) andfor an assemblyofgas-like particles (inwhich
the particles arefundamentally indistinguishable). Hence the statisticaldetailsfor
localized particles and for gases are treated below in separate chapters.

1.5.4 The average distribution

The reason for counting microstates is that, according to the postulate of equal proba-
bilityofallmicrostates, the numbert({nnj})isthestatisticalweightofthedistribution
{nnj}.Hence we can now in principle make the correct weighted average over all
possible distributions to determine the average distribution{nnj}av. And this aver-
agedistribution, accordingto our postulates,isthe one whichdescribes thethermal
equilibrium distribution.

1 .6 A model example

Beforefurtherdiscussion ofthe properties ofalargesystem, the realistic casein
thermodynamics, let us investigate the properties of a small model system usingthe
methodology of the previous section.

1.6.1 A simple assembly

The macrostate we consideris an assemblyofN=4distinguishable particles. We
labelthefour particles A, B, C andD. The totalenergyU= 4 ε,whereεis a constant
(whose value depends onV).